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Convex Spherical Mirror: Point Object Reflection and Descartes' Formula

Convex Spherical Mirror: Point Object Reflection and Descartes' Formula
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2015-02-05 11:55
AngleVectorGeometric ConstructionLocusGraphing
This GeoGebra example demonstrates the reflection of a point object on a convex spherical mirror, visualizing how light rays obey the law of reflection and how the image is formed. The core mathematical concepts include spherical geometry, reflection law, Descartes' sign convention for spherical surfaces (1/s + 1/s' = 2/R), and angle relationships between incident ray, normal, and reflected ray. The construction uses GeoGebra commands such as Point, Semicircle, Line, Tangent, OrthogonalLine, Angle, Rotate, Intersect, Vector, and Parabola to build the convex mirror (MirrorConvex), focal point F, directrix, incident rays, normals, reflected rays, incident angle θ_i, reflected angle θ_ir, and intersection points Q and A to trace the virtual image. This is suitable for high school physics optics or introductory college geometric optics, helping students understand virtual image formation by convex mirrors and the validity conditions of Descartes' formula.

GeoGebra Commands & Parameters

  1. Point(xAxis) → C
  2. Point(yAxis) → B
  3. Semicircle(B, C_1) → MirrorConvex
  4. Point(xAxis) → F
  5. Line(F, yAxis) → b
  6. Point(xAxis) → Directriz
  7. Line(Directriz, yAxis) → a
  8. Point(xAxis) → P
  9. Point(MirrorConvex) → A
  10. Line(P, A) → f_1
  11. Tangent(A, MirrorConvex) → h_1,
  12. OrthogonalLine(A, h_1) → i
  13. Point(i) → G
  14. Angle(P, A, G) → θ_i
  15. Rotate(G, θ_i, A) → G'
  16. Angle(G, A, G') → θ_ir
  17. Line(A, G') → d
  18. Intersect(d, xAxis) → Q
  19. OrthogonalLine(A, xAxis) → e
  20. Intersect(e, xAxis) → A_1
  21. Vector(A_1, A) → h
  22. OrthogonalLine(Q, xAxis) → g
  23. Intersect(MirrorConvex, xAxis, 1) → O
  24. Vector(P, A) → u
  25. Point(d) → D
  26. Vector(A, D) → v
  27. Vector(P, F) → w
  28. Vector(P, C) → c
  29. Parabola(F, a) → k
  30. Point(k) → J
  31. Line(J, xAxis) → j
  32. Tangent(J, k) → l,
  33. OrthogonalLine(J, l) → m
  34. Vector(J, F) → q_1
  35. Line(O, e) → f_2
  36. Point(f_2) → E
  37. Point(f_2) → K
  38. Point(f_2) → L
  39. Line(E, c) → n
  40. Line(K, xAxis) → p
  41. Line(L, xAxis) → q_2
  42. Intersect(g, n) → M
  43. Intersect(b, p) → N
  44. Intersect(q_2, yAxis) → R
  45. Vector(E, M) → q
  46. Vector(K, N) → f
  47. Vector(L, R) → r

AI Drawing Prompt Reference

Draw a convex spherical mirror as a semicircular arc with its vertex at the origin and center of curvature on the horizontal axis. Place an object point, a focal point, and a directrix point on the horizontal axis, and draw a vertical line through the directrix point as the directrix. From the object point, draw an incident ray to a point on the mirror, then construct the tangent and normal at that point. Mark the incident angle, and rotate the normal by that angle to obtain the reflected ray. Find the intersection of the reflected ray with the horizontal axis)Skip, and draw a vertical line through it. Also draw a parabola with focus at the focal point and directrix as given, take a point on it, and draw its tangent, normal, and a vector from that point to the focus. Finally, add auxiliary lines and mark relevant intersection points and vectors to illustrate the reflection and image formation.
The prompt above was compiled from the original drawing commands and is for reference only.